Sequential convergences on $MV$-algebras
Ján Jakubík · Czechoslovak Mathematical Journal · 1995
The notion of an MV-algebra was introduced by Chang [2].Various systems of axioms and various notation for MV-algebras have been applied; we shall use those from [4]; cf. also [13].We investigate sequential convergences on MV-algebras.The definition is analogous to that studied for lattice ordered groups (cf.[6], [8]), Boolean algebras [9], [11] or lattices [12].Let srf be an MV-algebra and let G be a lattice ordered group.We denote by Conv srf and Conv G the set of all sequential convergences on srf or on G, respectively.Next, let Conv b G be the set of all bounded sequential convergences on G; this notion has been dealt with in [10].All the sets Conv-s/, ConvG and Conv G are partially ordered by inclusion.Mundici [14] proved that for each MV-algebra srf there exists an abelian lattice ordered group G with a strong unit u such that srf can be constructed by means of G.In this construction, the underlying set A of srf is the interval [0,u] of G.We shall prove that the partially ordered set ConVeC/ is isomorphic to Conv b G. From this we deduce that each interval of Conv^ is a complete Bouwerian lattice.The lattice Conv^ has a greatest element if and only if ConvG has a greatest element.It will be shown that if [0,u] is a Boolean algebra, then the relation Conv srf = Conv £8 is valid (where £8 is the Boolean algebra under consideration, and Conv £8 is as in [9]).Supported by Grant GA SAV 1230/94